Wiles proved the semistable case of Taniyama-Shimura which was enough to do Fermat's last theorem Ribet showed that a proof exited for the full Taniyama-Shimura in 1999, finally being proved by Breuil, Conrad, Diamond & Taylor
so to your question
> In the broadest terms, would it be accurate to say that Andrew Wiles > succeed in _proving_ the Taniyama-Shimura Conjecture
>or did he only > prove, as Daney implies, a special _case_ of the Taniyama-Shimura?
yes, although 'only' is a probably not the right word!